Third Note on Weierstrass' Theory of Elliptic Functions

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Third Note on Weierstrass' Theory of Elliptic

Functions.

By A. L. Daniels, Johns Hopkins University.

The Sigma-Quotients.

As long as the argument and the quasi-periods 26), 26)', remain the same, we

S
may omit them, and write S, S l7 S 2 , S 3 , — - > etc. The functions S lt S 2 , S 3 are

then thus defined,

e-i»<5(«> + «) ei«S(a> — w)

GjM = -p = -p >

6<w So)

«-'»"« (3 («" + «) _ ei"" (3 («" — «)

Soit 3 T. 3 T, '

2 So/' Sft>"

er-i'« S (w' + w) ei'« S (a/ — w)
d Sft)' So)'

which are seen to be even functions. These apparently arbitrary definitions
flow naturally from considerations connected with the "pocket edition,"

S (u + v) S (u — v)

<pw—<pv= — — _,„ J- — •

S^M S*«

Since jpa = e lt p(a + <<>') = e 2 , #>6)' = e 3 , we have, using the general mark a, and

writing v = 6> a S (w + <"«) 6 ( M — "0

*"* — *•- S 8 «.SV

But pu is a truly periodic function : it remains therefore to examine the period-
icity of &u. In the second note, p. 261, I have shown that

6u = mH' w ( 1 — — J e- + 2 *»

«> = m.2o + m'26)'
degenerates into the sine when wi' = , or

hm (Sw) m , =0 =~e«^ ; .sin~.

Daniels : Third Note on Weierstrass' Theory of Elliptic Functions. 83

It was also shown that

The following definitions are introduced as convenient :

~- r logs(x) = -~=s 1 x = (-Y^ ( y~)

dx v ' s(x) x / ' \x — n nj

n = — oo
4-oo

d ^-n 1

~~ ~dx~- SlX ~ S * X ~ 2^(x — nY

in which last series the value n = is included. One sees that s % (x + l) = s 2 (x),
and s 2 x is periodic. Incidentally it may be remarked that on comparing the
developments for sin nx and s (x) ,

Sin 7tX = TtX 1 rr- + — ...

6\ 5!

" = -«=-(-(t)X-(t)0O-(t)0-

= x ~ + $ + & + * + • • O^ 3

)£C 5

-j-OO -f-06 -j~ 00 00 00 00

1 m=l » = 2 m = l «=2 j) = 3

m<^n m<Cn<Cp

whence v 1 _ w 2 vv 1 _ 7r 4

■^ 9 —~ m > ^^ 9~o" — - . > etc.
?r 3! bit 5!

We can now in - ■nv/'-i w \ - + 1-^

6m = «II' (1 Je" »«' ,

give to m' a constant value and carry out the multiplication with respect to vn .
For m' = , we have

/ u \ " -i. 1 ( u V

($u ■=. uli' ( 1 ) e m2w 2 v » 20 "'

\ wUcoJ

/ u \ w 1 ( u V

= Mil' ( 1 s- ) e m2w . lie 2 Vm2w ' •

\ m2a>/

Rut JL _L Hi I li

Xle 8 '"»*' ™ ! ' = e 8 6

and tt/T/. M \ -4-1 2w

wIT (1 =- 1 e m2w = — sm-i

L\ m2a)/ J k 2o)

84 Daniels : Third Note on Weierstrass' Theory of Elliptic Functions.

which two factors furnish that part of <5u corresponding to those values of m
and ml represented in the plane of complex number by points on the real axis
distant from each other by 2o ; in other words to the numbers

0, ±1.2o, ±2.2«, ±3.2©,...

Employing again m = ^ ^ _ j^ ^

we have s(u — a) u — a , / __ u \ !±

s ( — a) — a \ n -f- a)

Remarking now the identity

U MM M

n n-\-a n — n — a

and also s'j — a) 1 -, / 1 , _1\

s( — a) a \ — n — a nj

there appears s(u — a) _ / _ u\ n ,/ u_\ _«_ + i + _JL_ >

s( — a) \ a/ V M-f-a/

But s?(—a) , ,

therefore s(u — a) / u\ rT .r/ u \ —£-n «*,(-<*)+•—

—. r=( 1 ) n ' (1 ; — )e n + a \.e

s( — a) \ a / L\ n-\-a/ J

1 J e a into the product as the value of (l — —J e n + a for

n = , we can drop the accent of the product sign and write

s( "-" } = n(i--±-) e^r».e^-"\

s{ — a) \ n-f-ay

whereby the only restriction as to a is that it 'must not be an integer. Transposing,

\ n + a J s( — a)

We are now ready to decompose the sigma-product

n/ m \ " I 1 ( u V

m= 0, ± 1, ± 2, . . . ± oo,

m'= ±1, ±2, ±oo,

where »i' = is omitted, as already accounted for, and consequently the accent
on the product sign is dropped. Dividing by 2u lt the formula becomes

u

Hi i_ — ^__ | e ™ +m -: ie¥

m + m 1 —

\m + m' — /

Daniels : Third Note on Weierstrass' Theory of Elliptic Functions. 85

Writing as before

S 2 (») = -5— «!» = S -7 " To '

dx

the second exponential factor becomes

1 (n_y „ 1

V2a>/

(n — xf '

for each particular value of to', the summation being taken with respect to m
alone. The rest of the product is

u

2w

n i

m + m'

7

'(-•"'£>

by the formula above deduced. Collecting the four factors, we have

_1 f^ «2 / w \ "*"°°

<3'« = e» • e -»» g f ) # n

This formula can however be simplified in form by multiplying together the
factors in pairs and taking the product from 1 to a>, instead of from — oo to
+ oo. For we had +co

s (x) = xTL' (l )e n >

1 + 00 / 1 1 \

. 1 (- x ) = -±+ 2' (— — +-)

_„ (a; + w) 8

+<» 2a;

whence it appears that s x ( — x) = 2 j j> or is an odd function, consequently

-4-oo -f"°° -f-°°

and one exponential factor disappears, and the formula now reads

6u — 2oe 8

f u i w '\ f u I > 0J '\

*" m ' =1 s (~ m '^)- s ( m '^)

= 2oe^T^ s " gSTVP + liJ^M "="> n

2w

M jr TO S[ h TO

/ u , a>'\ / u , «/\

si — m — J s ( to — J
Vol. vii.

86 Daniels : Third Note on Weierstrass' Theory of Elliptic Functions.

and, on passing from the s and s 9 to the sine,

it Tt (y^Y

o i /<r«\s sm h~ (2mV — u) sin — (2m'a' + u) w»{, ,

(5M=-sin— . e 6 V2 " ,/ II r-j . e sm "^r" .

n 2cd . 9 m'(o'n

(!)

Professor Schwarz writes

' 9„. -/ft T -^ . . rn'fo'v V

2« -< 6

^ sm

whereupon the sigma-function is thus represented as a singly infinite product of
sines / ^f^L}

6u = e*>° . — • sm — — n '

n " 2co 1 i.A . „/mW7r'

On substituting w + 2a for m the expression becomes

6{u + 2a) = — e 2 " ( " +<o) (3w,
from which by logarithmic differentiation and writing u = — a, we find

<3m

Recurring now to the pocket edition

6(u-\-co a )6(u — co a )

From the definitions at the beginning of this paper

6 2 zt = e 2 "« tt — i-= ,

so that /6«w\ 8

which is the simplest form of a doubly periodic function. In the second note
was deduced the equation {tfuf = 4 (pu — e^)(pu — e^)(pu — e 3 ) ,
and on comparison with the above

6u.6u.6u
The sigma-quotients have not the same pair of fundamental periods as the
sigma-function itself. But while

6u has the quasi-periods 2a, 2a'

-4— has the periods 2a, 4a'

^ " " 4a, 2a'

<ou

^ « « 4a, 4a'.

6u

Daniels: Third Note on Weierstrass* Theory of Elliptic Functions. 87

This is shown in the following manner. It will be noticed that aside from

exponential and constant factors, the function 6 U G> 2 , <S 3 , are formed from <ou by

increasing the argument u by the half-periods a, o" = &) + &>', J, respectively.

If we write w = no + r'o', V = nq + r'rf

instead of w = m2a + m'2d,

it is evident that w and w will only then be equivalent when both r and / are

even. We have then

<o(u+ 2w) = £6u.e*> (u+ *\
<o(u + to) = e<o(u — w)e^ = — s6(w—u)^ u ,
and, writing u = 0, <5w = — e(5(w>), orf = — 1 when either r or / is odd. If
both are even, then w = w and <5w= 0. To determine the value of e in this
case, develop both sides according to powers of u

u.<S'w + u 2 + . . . = e.u.&w + . . .
and s = + 1 when both r and ?•' are even. Now the formula

(r+ 1)(/+ 1)— l = rt J + r + /
is only even when both r and r' are even ; we can write therefore,
<5(« + 2w) = (— l) rp ' +r+r '. (3m. <?«•+«
<3(w + o a + 2w) = (— l) rr ' +r+r '.6(u + G> a )e SH > (M+lu «+ W) ;
but, from the definition (3(w + cj a ) = e" a *.0vw.6G) a ,
whence, writing for u, u-\- 2w

6 (u + o a + 5E5) = e*« ( " + m <o a (u + 2«o) . 6a a ,
and, equating the right-hand members,

<5.(w + 2w)<5co a = (— l) rr '+ r+r '.<3(-M + 6) J e ^ (M + ' *+ i2) -''« (tt +»
or, writing u — w for u

& a (u + u-)Qa a ,= (— l) rr ' +r+r '.(o .(u — w)e 2i7i "'-- ri - :S) + ^ w ,

6 a (u-\-w) — . iW+r + f'JS.,-,,S) + «i»

(S a { u — w)~ \ L)
and for u = , since (5 a ( — w) = <S a ( + w) we have for the determination of r and /,

For the case a= 1, we shall have yj — rrj + r'r/, w = no + r'a 1 , vj a =yi, a> a =-a,

an( ] J = gJr'tu'" — v»')+ («•' + r + r')iri

But , , id

r\ o — >7« = zfc — >
A

whence ^ (w + 2w) = (— 1 ) rr ' + r . 6 x u .e**(u + w).

For the case a = 3 , we shall have

>? a = V, "a = »', 2 (rw a — »y a w) = 2r (r t d — j/o) ,

88 Daniels : Third Note an Weierstrass' Theory of Elliptic Functions.

and <o 3 (u + 2w) = (— l) rr '+ r 'S 3 M.e 2i,(w + w ,

and likewise 6 2 (u + 2w) — (— l) rr '<3 % u<?' n{n+ ™ ) ;

so that 6 1 (u + 2w) __ _ ._ y r , +r 6&

6{u-\-2w)~ \ ' ' 6u'

6*(u + 2w) __ _, iyK+K <^u,

6(u + 2w) l ' ' 6u'

6 2 (u + 2w) _ _ , y r , 62U,

(5(u + 2w) ^ ' ' <5u '

In order therefore that 2w= 2 (ra + rVo') may be a period of -J- > we must have
(_ j^r'+r+i— j ( or rr >_|_ r _|_ 1 — even, r(r'+ 1) = odd, that is ?•= odd, /= even,
so that 2w = 2mo + 4m'6>', where in and wi' are integers. In like manner, for
a=3we shall have 2w = 4ma> + 2m' J, and for a = 2 , 2w = 4mo + 4mV.

The relation will now be shown between the sigma-quotients on the one
hand, and the notation of Jacobi and Abel on the other. The Jacobian differ-
ential equation is £fcy = (J _ ^ _ ^

In the second note, p. 267, we had

(p'uf = 4(pu — ej)(pu — e 2 )(pu — e s ) ,

. n pu — e K =(~ S );^=l,2,S,

or, since r A \6uJ ' ' ' '

q'u = — 2 * •

<ou.Qu.Qu

Writing now for convenience p— = £ 0A , -A- = £ M> ,, etc., the last equation beco:

6 K u ^ K1 (6 v u

For « = these functions £ satisfy the conditions

£<w = , £„„ = 1 , £ A0 = 00

From

p«— e A = f p^-J, 3,= 1, 2, 3, = 3,, (U, v,

we obtain 6£ -« — <S\ u + (e M — e„) 6 2 w = ,

& v u — 6\u + (<?„ — e K ) &u=0,

<5\u - 6> + \e K — ej S 2 w = ,

(e„ — e v ) 6 x u + (e„ — e A ) 6 m m + (e A — <? M )6„w = .

The differential equations are then thus transformed

\duj~ \du ($J ~ «»*■«•*— <5»<3»
~~ 61.(61

Daniels : Third Note on Weierstrass' Theory of Elliptic Functions. 89

(^H i -fe-<0][i-(--<0].

and similarly (±.^= p -)».][,._*+ (*-*)&],

and, in general, the four functions

6u

6m

6 K u

satisfy the same differential equation

(ID = C 1 - fc-OPKi - («,-*)?)■

In order to compare these with the Jacobian differential equation, we have only
to write

^x— «v£oa=£, % = Ve A — e M .w,

<V «A

whereupon

<V — «a

= 7^

V«A 6 W

e 6u sn«! sn(\/e\ — e^.Ujh)

<5 A w \Ze A — <V Vex — 6 M

and in a similar manner all the twelve sigma-quotients are produced,

6u 1 , , 7 .

-^- = —p sn (ve, — eo.u, k)

— =cn(V ei — e 3 .u,k)

6 2 u , . , TN

— =dn(Ve 1 — e 3 .u, 7c)

6 X M

= sn coam (Ve 1 — e 3 .u, k)

-zr—= —j cos coam (Ve x — e s .u, h)

6 8 w \/fii — <%

6 lU _

en (v^i — e s .u, k)
sn (V^i — «3-M, A)
^2 W _ / dnjy^ — 63. u, h)

<ou 1 3 sn (\^e l — e 3 .u, k)

63U
<5u

= Vej-

! sn(V«i — ea.u, k)

-=— = , tn ( Ve, — e, . tt , 7tf)

<3 2 w 1

£>!« sin coam ( \/«i — "3 • « > A)

<3 3 w 1

6 x m en ( \Z«i — 63 • w > &)

== A coam (Ve, — f, . u , A;)

coam (Vej — e 3 .w, A;) = am (K — s/e x — e s .u, h).

Abel writes (Oeuvres, t. I, p. 265, nouvelle edition),

_ r dx

V(l — cV)(l+eV)'

x = <pu, */l — c*aj* =fu, Vl + e 2 « 2 = Fu,

90

Daniels : Third Note on Weierstrass' Theory of Elliptic Functions.

comparing which with the* Weierstrassian notation,

Fu —

(3m . 6,u

x =■ <pu ■=■ —r- i tu= ^

6 s u

if only e 1 — e % = — c 2 , — (e 3 — e 2 ) = e 2 .

As regards the analogues of Jacobi's K and K , it is to be noticed that, as
usually defined by the equations

K - f g K - r * _

the values are only unambiguous when the path of integration is fixed, it being
generally understood that the path of integration is the straight line from to 1.
Corresponding to this we have, e. g.,

K = s/e x — e 3 (a + 4p« + 2qa') ,
where the determination of the path of integration corresponds to the freedom
of choice of p and q. Commonly we have p = q = 0, and

K= aVe 1 — e 3 , iT'i = co'-v^

e 3!

and then 2a, 2a' form a primitive period-pair for the function <p(u, g l , g 2 ), and,
if we write as before a + a' = a", or ^ + a 3 = o 2 , then is fa x = e^ pa^ = e 2 ,

The functions G^w, 6 2 m, <5 3 w, can be represented as an infinite product of the
same form as that for 6u by writing

w.

(2(i+l)a+2(i'a', w 2 = (2 w + 1)u + (2m'+1)g)', tu 3 =2 W 6) + (2M+l)o',

where ,a, ,a'= 0, ± 1, =b 2, . . . ± 00 ; namely,

6 A w = e~ ie ^TlwA 1 ) e«>A + 2 «*.

But these functions are also representible in the form of singly infinite products.
As an aid in transforming, Professor Schwarz makes use of the following table.
When the argument u assumes the values u + a , u + a', n + a", then the magni-
tudes -»=:-—, a = e mi , 2yiai?, ^^ assume the values in the table, where t = — >
h — e™\

u

w + «

M+ «'

U-\- w"

V

v + i

-y + Jr

-y+i+l'-

z

iz

Tliz

j.7l.*.2

%nw?

2/?uW 2 + TO+^u

2<?«v 2 + V« + JV«' + \ttti -\- vni

S^uw' 2 + ?/"tt + 2 1 ?""" + jfi + ii"7ri + vri

2t)0)1>2

e 2ij<oD 2 _ g r|tt gsl"

J^^'u _ e hV h i z

<? VM '\e 7 >" u .e* ri "' ".Ji.hKz

Daniels: Third Note on Weierstrass' Theory of Elliptic Functions. 91

With the help of this table the infinite product for 6u on page 261, Vol. VI of
this Journal, can be transformed as follows : Developing

(1 — h 2 \^)(l — h^z-*) = 1 — 2A 2 "cos W - + h in

since

un n . „ un
cos — = 2 sur -tr-
ot 2(o

un

(1 — £ 2 V)(1 — tf n z~*) = 1 — 2#" + __V". sin 3 — + h

in

un

fan ft— » n(0 l

but — ~ — = i sin n — > consequently

= (l — h-Y+4Jt*.sm i -

sm 2 — J- ,
2<u

(1 — A»V)(_ — #•«-») = (1 — A 2 ") 2

sin"

M7T

~2w

sm'H

and

2w _ 2 — 2-1 1 — A 2 V 1 — /
n 2. "»1 — A* 1 — A*

which is the desired expression for 6m. Since further

<5> = e ~ vu • v ^ y ,
<5ft>

we obtain by the assistance of the table the analogous expressions for <a lt 6 3 , 6 3 ;

namely ,_■ »„„-, „ cos(tt7r — t>)7r __. _ cos (nt -{- v) it
J Qm = e ir "" v . cos vnTl- — . e vn . I_„ — ! — —

cos nm

cos nzn

. e"

,« z + z 1

?i«w* . ~ ! ~ _ n *

" i+a 2m

.n,

i + '

i+ft 2n

^e^.coswi.n

l + 2ft 2ra cos2iOT-fft 4 "

(1 + A 2 ") 2
.e~

& M =^-.iL ooB((n -* ) :r <,)ar .^-" < .n, C0B((n ~ i)r+ ^ -<

cos (n — J) rs-

1 I &2»-l --2 1 I /,2ra-l 2 2

— ^« 2 n — t__ tt +

cos (n — ^) rs-

-.e"

l » 1_A 3 »-

1 + A 2 "" 1

_ j*». tt l + 2A 2 "- 1 .cos2^ + /^- 2

6 3 u=e %vm \Yl,
= e 2r " M \U,
= e 2 ' ,w1 'Ml,

(1 + A 2 "- 1 ) 2
sin ((n — J ) t — v) n Bin((n — j)r + _ ) jt {

.e~

Mi

-1-2

sin (n — |) T7T

1— ft 2 "- 1 ?- 2 n

1 _ 2ft 2 "- 1 . cos 2wr + A 4 "- 2
_ __ _____

n sin (tt — i)™

-.e"

92 Daniels : Third Note on Weierstrass' Theory of Elliptic Functions.

Analogous to the expression for <5u at the bottom of p. 261, we have

. , un

<o x u = e 3w . cos -z— . IL.
2a>

COS* TO

. 9 WT

<5 s M=e» M .nJ 1 —

cos 3 (n — -§•)

<5,m = e 2<0 .n.

w /
. „ wr \

a/7r

sin 2 ^-!-)-—

In the normal case we shall have a real and — imaginary, and therefore
none of the quotients under the product sign can assume the value unity. The
functions <ou and 6 x u disappear accordingly only when sin — — and cos — —

respectively vanish. The Jaeobian functions -=- » -—- > -A » are analogous, the first

to the sine, the second to the cosine, while the third remains positive for real

values of u. The Abelian forms —r-> —■, -£~> are analogous, the first to the

6 3 6 2 6a

tangent, the second and third to the secant.

The expressions for the root-differences and the connection with the S-func-

tions are obtained in the following manner. Defining as above h = e a = e T1Ti ,
and writing

h = n (i — h* n ), h 1= n (i + a 2 "), h 2 = n (1 + a 2 "- 1 ), h s = n (1 — ^- 1 ),

then is A = A . ^ . A 2 . A 3 ; h t . h % . h s = 1 .

For fl« = (1 — # )(1 — A 4 )(l — A 6 ) • • •

= (1+A)(1+/* 2 )(1+F)...
. (1 — A)(l - h?)(l - A 3 ) . . .
and the proof is apparent. With the aid of these facts the relation between the
periods and the root-differences is easily discovered. Starting again from the

" pocket-edition " 6 (u -f v) 6(v — u)

pu pv — -^ ^o )

and writing u = a , v = a', a"=o-\-al, the equation becomes

<oco".<o(co' CI))

e, — e s = — _„ ' .

Daniels: Third Note cm Weierstrass' Theory of Elliptic Functions. 93

But noticing that o>' — a = o" — 2a, a" — a = a', and that

6 (a"— 2a) = — 6 (— a" + 2a) = — <3o>". e~ 3 " (M '-' u) ,
6 (<y_ o) = (5 («"— 2a) = — (36)".e- 2 " ( ^- w) ,
the equation becomes

And in a similar manner, writing u = a , t? = o> + a', we have,

1 2 ~ 6 3 «><SV ~~ \<o(o<oat'J '

and for w = o", « = «',

(5> + 2 t ,/')(3» _ / GW Y M"'

2 3— SV6V ~~ \6a>6w"J *
But these formulas can be still further simplified. From

we have for u =

a

2m 5» 2 un> ^1— A 3 V 1 — h 2n z~'

(5?/ ^Z fi*«) OTl I I .

(3o) = — e 2 II — ! — = e 2 —

And similarly & , _ ,£ _1_ a~* — ■ ^ >

The expressions for the root-differences become then

where h = e"™"" , and ^ , Aj, A 2 , h 3 , are defined above. In accordance also with
previous definitions for the Jc and W of Jacobi,

jA-^z^i- ifi* f (i + ^(1 + ^X1 + ^ ••• •)"
«i-^ ~ 1 (i + A)(i + A 8 xi+^ 5 ). • J

in-^ZZ^- ( (l-h)(l-h*)(l-h*)... y

e.-e,- 1(1 + A)(1+A 8 )(1+A 5 )...J '

The four sigma-functions are now expressed through the functions 3 and © , as

follows. The infinite product F(z) =U n (1 — A 2n )(l + A 2m - 1 2- 2 )(1 + A 2w - 1 3+ 2 ) can

be expressed as a power series of z 2 which converges for all values of z except

Vol. VII.

94 Daniels : Third Note on Weiemtrass' Theory of Elliptic Functions.

z = 0, so long as h<C 1. This is plain from the following identity,

n m (i — ¥ n )(l + h^-H-'Xi + A 2 "-V)

= 1 + h (z 2 + %~ % ) + A 4 (z 4 + sT 4 ) + A 9 (z 6 + z~ 6 ) + ...

The development of the sigma-function follows at once, since

— /2co

hl.h .6< i u = e io ' .F{z) and^y — \Z^ — e% = h J%.

We have then

^J™ 4/^7 8 . <6 % u = e^F (a) = e^ . 2 W

^/^ ^Z^,. <5 3 « = e^ F(zi) = e^.X (— 1)"A"V»

/o77 i u * i" 2

v /f f0 ^ZT^. (S lU = e^. h*.z.F(zh*) = e^ . 2#<*«+»y»+ 1

/2^ 1 W* ,

>/ — v'e! — e 3 v'ej — <v a/^ — (%. 6w = -« h h*z .F (izh?) .

It will be remembered that e x , 6>, e 3 are the roots of the equation

4a; 3 — g2x — g 3 = .
The discriminant of which, squared

\(ei-e,)(e 2 -e 3 )(e 3 -e 2 )r=^ 7 ^= G,

so that /"9^T 1 ^ i" 2 1

W— • ¥G = 4- e^tt.z.Fihh) = e*±X(— iyh i&n +^z 2n+1

« = 0, ±1, ± 2, ... ± oo,

M7t£ to/ ,
— - .TTl

The expression for .Fz becomes, since

^+ z -^=2cos~.

CO

F(z) = 1 + 2h cos h 2A 4 cos 2 \- 2A 9 eos 3 f- . . .

W CO CO CO

which is the 3 series of Jacobi (Werke, Bd. I, p. 501). Weierstrass defines the
3 functions as follows :

T S(— i)»Ai(2»+i)= z 3»+i— 2A*sin wt — 2A*sin 3wt + 2^"sin 5w= . . . = ^(w),
2A i(2w+1) V a+1 = 2**008 wi + 2A* cos 3wi + 2A^cos Son + . . . = 3»(w),
2 fc»V» = l + 2A cos 2wt + 2£ 4 cos \mt + 2W cos 6vn + . . . = $ 3 (w) ,

2 (— l) w A w V n = 1 — 2h cos 2V71+ 2A 4 cos 4wt— 2A 9 cos 6wt + . . . = 3 («) ,

& »

Daniels : Third Note on Weierstrass' Theory of Elliptic Functions. 95

which agree with Jacobi's notation when vn = x ; h = q. Hermite writes

m= 0, ±1, ± 2, ... ± oo.
If now we define the function @ by the equation

©„ (u) = e*"". 3„ (v , — ) ; u = 2cw ,

then the four sigma-functions are thus expressed through © ,

V'^r ^^v^^^'s,,^, ~) = e,(u, a,, «'),

where again « = — — ■• By developing according to powers of v, and comparing

we have /^ = 1 = jr_^ (1 _ 3 ^. a + gA „__ 7F 4 + _ }

^ n 2(0 K to K '

s/^- vV=^=3 2 (0) = 2tf(l + #•»+&« + 7*" + • • •)
n/-^- jfo^ = 3,(0) = 1 + 2A + 2A 4 + 2A 9 + . . .

^/^ 4/eT=^= 3 (0) = 1 — 2A + 27i 4 — 2A» + . . .

From these spring the following equations which become useful in computation :
3^(0) = 7i^ (0).3 2 (0).3 3 (0), 3 4 (0) +3|(0) = 3|(0),

^g = «* + a tf + «¥ + ... = 2 (2A+2/ , + 27j25+ .., )

^ = i+»+^±y+- = » — (i + MH- «. + .■■)

('•3

_ #«„. — «. _ °*\ ' coj _ 2M + 2M + 2^+...
^ - ^^=^ ~ I / A <A ~ l + 2A + 2^+2A 9 + . . .

3

«)

96 Daniels: Third Note on Weierstrass 1 Theory of Elliptic Functions.

v# =

jfo=j, ~ . /- a/\ 1 + 2h + 2h* + 1h? + . . .

»>«)

We define Q /_ , at

,_ 1— V^_ #<fc — «r— v^a — <V_ 2/i-f2/i 9 + .

3.

1 + V*' V e A — e, + #e K - e» 1 + 2A* + 24 16 +
which is identically satisfied by writing

*= 3 =4+ <4) s + K4)'+ "o({)"+. . .

These expressions for I and A make the computation of the period 2o or of
Jacobi's K very easy.

In explaining more at length the methods employed for computing, I cannot
do better than to give them with scarcely any variation from the words of that
most genial expounder of Weierstrass' theories, Prof. Schwarz. When the three
roots <?i, a>, e 3 , are once known, we can, by the aid of the formulas just given,
not only compute with the greatest ease the two periods 2o, 26)', but we can
also express the sigma-quotients through such 3- series that the argument h shall
have the smallest possible value, and the series converge most rapidly. This
last end is brought about by so choosing the order of magnitude of e x , e 2 , e 3 , that

, */e k — e„ — y/e K — <v

which is used in the computation of h shall be as small as possible. Of the
several cases which present themselves according as the invariants g%, g 3 and
the roots e 1 , e 2 , e 3 , are real or imaginary, I shall discuss here but one, where all
are real. The roots will be real when the discriminant G = -% {g\ — 27#|) of
the cubic equation 4s 3 — </ 2 s — g 3 = is positive and g % and g 3 real. We then
assume e 1 >e 2 >e 3 and all the radicals positive; farther /l=l, [i=2, v = S,
whereupon

■=Vi, -Z7T=Vt, *= — > h = e™\ h 1 = h'=e ',

(6w 1 " 1I 6w s "*' ' CO,

u ui , co s r , . .

v = — — 1 v, = - — 1 where 6),, — > -r-i h x h, are positive.
2(o t 2(o s 1 %

For the computation of the periods the following system of equations is used :

m^M' *.=4 + <4)"+ »&)'+ -(4)"+ ■ ■ •

l =

Daniels : Third Note on Weierstrass 1 Theory of Elliptic Functions. 97

V 7 — = T7 _f tj— (1 + 2A 4 + 2A" + . . .), %= ^ log. nat. -J-,

?r s 1 — 3 8 ft 3 +5 s ft 6 —7 3 ft 1!! + . . . _! .

2*7i<»i--g- 1 _ 3ft8+5ft 6_ 7ftl2 _ J _ -i> ' >7i% — toi*!,-*™,

s /^V~^=^=2tf(l+h?+h«+h 1 * + .-.)
^/^°L^^zr^= 1 + 2^ + 27i 4 + 27i 9 + . . .

2c0i 4

Ve t — 63 = 1 — 2h + 27i 4 — 27i 9 + . . .

N /^^ = --A { (l- 37> 2 +57i 6 — 7A 9 + . ..)
For the calculation of the sigma-functions we shall have

\/^ V^=^. (S 3 u = e*-" S (« , r)

3 (w, t) = 1 — 2A cos 2vn + 27i 4 cos 4wt — 2W cos 6vn + . . .
3>! (v , t) = 2/i ¥ sin wt — 27t* sin 3wt + 2/^ s [ n § V7l — , , .
3,, (v , r) = 2h* cos wt + 2/i* cos Svn + 2A* cos 5 vn + . . .
S. g («, t) = 1 + 27i cos 2oti + 27< 4 cos Amt + 27t 9 cos 6vn + . . .
If, however, we have to choose 1 = 3, p= 2, v = 1 , the equations become

^ ^-^ Ai= |_ + 2 /|v + 16 / iy + ir / <m»

150

+ ...

m Vej — es+V^a — e s v ' m & Vfti/

v/^r ■ v%=^= 274(1 + A? + AJ + A? + . . .)
x /^.^/^r7 3 = 1 + 2/^+ 2/i 4 + 27i? + . . .
^/^,^/-^ZZe s = l — 2h 1 +2hi~2h\+. . .

x/^ • VG = — ^(1 — 3h\ + 5AJ— 77>.f + . . .)

OT

98 Daniels : Third Note on Weierstrass 1 Theory of Elliptic Functions.

^ TTl % \ T J

sj^ ^^=^.6 3 u = e-^^l^(y 1 i, =1).
$o («i» . = p) = 1 — Ai (e 2 " 1 ' + e- 2 "" 7 ) + hf (e 4 »>* + e" 4 "^) — h\ e^ + e- 6 "'") + . . .
■i-^i («i*f = ^r) = A*(e""— e - " 1 ") — A* (e 88 -— e 3 ^) + h?(J" w — <r*"') — . . .
S 8 («i* f = ^) = A* (<?«' + e-^) + Af(e 3, '" r + e""'') + A* (<*•»* + e- 5 ""') + . . .
S 3 («!*, — ) = 1 + Me 2 "'" + <*-*"•) + A| (<**•" + e- 4 "'") + hl(e tv >" + e~^) + . . .
When e % — e 3 = e 1 — e 9 , that is, e 2 = 0, then is

Z^fj- 1 , H,ir' and^fjr^, A^tf-

When # 3 is positive, that is e 2 <C 0, it is advisable to use the formulas for h and I,
otherwise those for A x , 4 will be found best, because in the first case we have
h<Ch lt in the second hx<lh.

For the calculation of the elliptic integral of the first kind in the ordinary
form, Professor Schwarz throws the necessary formulas into the following shape
"Among the values of u for which <pu-=.s, there are, in consequence of the
equation 6 8 (ti± 2a/) 6 s u

6 2 («±2a/) 6 3 M

4/

always such for which the real component of -; . x is not negative and conse-

Vei— eg

quently the modulus of

\/ \ — e s .<o 2 u — 4/ \ — e 2 .6 3 ?,f \/e t — ^ — v^ — e% G^u, w, 4a/)

v'ei — eg.(5 2 M + \/e t — e 3 .6 3 w v^ — e 8 + 4/^ — ^ * 6 3 (2m, w, 4«/)

y

is not greater than unity. The value of \/s — % can be chosen at pleasure, after

which the value of Vs — 63 can be so taken that the real component of

Daniels: Third Note on Weierstrass' Tfieory of Elliptic Functions. 99

shall not be negative. We then write

+/ \ — e 3 — V«i — e z 7 Ve t — e 3 - Vs — eg — V^i — e 2 .*/s — e 8 ,

a/«i — «3 + ^«i — «2 ' ^^ — eg./n/s — % + -\/e t — e 3 .\/s — 63
L = 1,

^= 1+ (1) V '

X 2 = 1 + (1)V + (1:

1 ^

when the equation

2 r 1 dt

(t/«i— % + \/ei— <h)'

Jx. y log. nat. (< + iVl^^ 2 ) + VI -^[^-^(ft)

+4 ^W+ It- ^W+ ■•■]}■

determines such a value for u as satisfies the equation pu = s, when to Vl — ^ is
given either one of its two values, and to the log. nat. any one of its infinite
values."

It is to be hoped that notwithstanding a few gaps in the demonstrations,
this sketch has been elaborate enough to give mathematical students a clear idea
of these theories in themselves and in relation to the older nomenclature of
Jacobi.